Adaptive Gaussian Mixture Filter Based on Statistical Linearization

Gaussian mixtures are a common density representation in nonlinear, non-Gaussian Bayesian state estimation. Selecting an appropriate number of Gaussian components, however, is difficult as one has to trade of computational complexity against estimation accuracy. In this paper, an adaptive Gaussian mixture filter based on statistical linearization is proposed. Depending on the nonlinearity of the considered estimation problem, this filter dynamically increases the number of components via splitting. For this purpose, a measure is introduced that allows for quantifying the locally induced linearization error at each Gaussian mixture component. The deviation between the nonlinear and the linearized state space model is evaluated for determining the splitting direction. The proposed approach is not restricted to a specific statistical linearization method. Simulations show the superior estimation performance compared to related approaches and common filtering algorithms.

💡 Research Summary

The paper addresses a fundamental challenge in nonlinear, non‑Gaussian Bayesian state estimation: how to choose an appropriate number of Gaussian components in a Gaussian‑Mixture (GM) representation without incurring prohibitive computational cost. Traditional GM filters either fix the number of components a priori or rely on ad‑hoc merging and pruning strategies after the fact. Both approaches suffer from a trade‑off—few components lead to poor approximation of the true posterior, while many components cause an explosion in runtime.

To overcome this dilemma, the authors propose an Adaptive Gaussian‑Mixture Filter (AGMF) that dynamically adjusts the mixture size based on a quantitative measure of the local linearization error. The method is built around statistical linearization, a family of techniques (Unscented Transform, Cubature Kalman Filter, probabilistic first‑order Taylor expansion, etc.) that approximate a nonlinear transformation by matching moments of a set of weighted sigma points. For each Gaussian component (i) with mean (\mu_i) and covariance (\Sigma_i), the filter computes an error metric

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